Observations on power series involving logarithmic singularities

[jaydfox]

To illustrate my point a little better, I put together the following comparison of the power series of sexp(z-1) and the power series of ln(z+1):

Code:

|  n |       sexp(z-1)      |       ln(z+1)        |       difference
|----+----------------------+----------------------+----------------------
|  0 |   0.000000000000000  |   0.000000000000000  |   0.000000000000000
|  1 |   1.091767351258320  |   1.000000000000000  |   0.091767351258321
|  2 |  -0.324494761735111  |  -0.500000000000000  |   0.175505238264889
|  3 |   0.349836269767157  |   0.333333333333333  |   0.016502936433824
|  4 |  -0.230854426837443  |  -0.250000000000000  |   0.019145573162557
|  5 |   0.201330212284523  |   0.200000000000000  |   0.001330212284523
|  6 |  -0.164352165253219  |  -0.166666666666667  |   0.002314501413448
|  7 |   0.142836335724572  |   0.142857142857143  |  -0.000020807132570
|  8 |  -0.124694993215245  |  -0.125000000000000  |   0.000305006784755
|  9 |   0.111073542269792  |   0.111111111111111  |  -0.000037568841319
| 10 |  -0.099954567162944  |  -0.100000000000000  |   0.000045432837056
| 11 |   0.090897908329423  |   0.090909090909091  |  -0.000011182579668
| 12 |  -0.083325611455161  |  -0.083333333333333  |   0.000007721878172
| 13 |   0.076920407600429  |   0.076923076923077  |  -0.000002669322648
| 14 |  -0.071427110437354  |  -0.071428571428571  |   0.000001460991218
| 15 |   0.066666069650214  |   0.066666666666667  |  -0.000000597016452
| 16 |  -0.062499703108237  |  -0.062500000000000  |   0.000000296891763
| 17 |   0.058823398084957  |   0.058823529411765  |  -0.000000131326808
| 18 |  -0.055555492550941  |  -0.055555555555556  |   0.000000063004615
| 19 |   0.052631550022682  |   0.052631578947368  |  -0.000000028924687
| 20 |  -0.049999986272057  |  -0.050000000000000  |   0.000000013727943
| 21 |   0.047619041200907  |   0.047619047619048  |  -0.000000006418141
| 22 |  -0.045454542411656  |  -0.045454545454546  |   0.000000003042890
| 23 |   0.043478259432923  |   0.043478260869565  |  -0.000000001436642
| 24 |  -0.041666665984126  |  -0.041666666666667  |   0.000000000682540
| 25 |   0.039999999675934  |   0.040000000000000  |  -0.000000000324066

Edit: always got confused on how to write a shifted series. If it's the sexp centered at -1, it's not sexp(z-(-1)) or sexp(z+1), it's just sexp(z-1). Arrgh! Likewise, ln centered at 1 is ln(z+1).